COS3761
Assignment 3
Formal & Module Logic
Due 2025
,COS3701 Assignment 3: Formal Logic 3
Question 1
In which world is the formula ♢p ∧ □q true?
Answer: Worlds x1 and x3 .
Justification: In x1 , the only accessible world is x2 , where both p and q are true. Hence
♢p and □q both hold at x1 . Similarly, from x3 , the only accessible world is again x2 , so
both ♢p and □q are satisfied.
Question 2
Which of the following does not hold among the given modal evaluations in the model?
Answer: x1 ⊭ ♢♢p
Justification: At x1 , the only accessible world is x2 , and x2 has no accessible worlds.
Thus ♢p fails at x2 , and therefore ♢♢p fails at x1 .
Question 3
Which of the following modal statements is satisfied in the model?
Answer: x1 ⊨ □p
Justification: All accessible worlds from x1 (namely x2 ) satisfy p. Therefore, □p holds
at x1 .
Question 4
Which of the following formulas is true throughout the entire model?
Answer: □q
Justification: At every world, all accessible successors satisfy q, and in worlds with no
successors, □q holds vacuously. Hence □q is true at all worlds.
Question 5
Which of the following formulas is false at some point in the model?
Answer: □♢p
Justification: At x1 , the accessible world x2 has no successors, so ♢p is false at x2 .
Hence □♢p fails at x1 .
1
, Question 6
Rewrite the English sentence: “It ought to be the case that if it rains outside then it is
permitted to take leave from work” into modal logic using p: ”It rains outside” and q:
”Take leave from work”, with permission formalized as ¬□¬q.
Answer:
□(p → ¬□¬q)
Justification: □ϕ is interpreted as “It ought to be that ϕ.” Permission to q becomes
¬□¬q. Therefore, the conditional obligation becomes □(p → ¬□¬q).
Question 7
Explain why the modal schema □ϕ → □□ϕ corresponds to axiom 4 of modal logic.
Answer: This is axiom 4 of modal logic.
Justification: If ϕ is necessarily true, then it is necessarily necessary. This requires
transitive accessibility relations and ensures that if ϕ holds in all reachable worlds, then
it holds in all worlds reachable from those as well.
Question 8
Under the epistemic interpretation of modal operators (where □ϕ means ”the agent knows
ϕ”), what does the schema □ϕ → □□ϕ express informally?
Answer: If the agent knows ϕ, then the agent knows that they know ϕ.
Justification: This is positive epistemic introspection. It reflects a principle of rational
knowledge that one is aware of their own knowledge.
Question 9
Which of the following modal formulas is not valid in the basic modal system K (nor in
the T–K–4 fragments):
a) □p → p
b) □p ∨ □¬p
c) □p → ♢p
d) ♢p → □♢p
2
Assignment 3
Formal & Module Logic
Due 2025
,COS3701 Assignment 3: Formal Logic 3
Question 1
In which world is the formula ♢p ∧ □q true?
Answer: Worlds x1 and x3 .
Justification: In x1 , the only accessible world is x2 , where both p and q are true. Hence
♢p and □q both hold at x1 . Similarly, from x3 , the only accessible world is again x2 , so
both ♢p and □q are satisfied.
Question 2
Which of the following does not hold among the given modal evaluations in the model?
Answer: x1 ⊭ ♢♢p
Justification: At x1 , the only accessible world is x2 , and x2 has no accessible worlds.
Thus ♢p fails at x2 , and therefore ♢♢p fails at x1 .
Question 3
Which of the following modal statements is satisfied in the model?
Answer: x1 ⊨ □p
Justification: All accessible worlds from x1 (namely x2 ) satisfy p. Therefore, □p holds
at x1 .
Question 4
Which of the following formulas is true throughout the entire model?
Answer: □q
Justification: At every world, all accessible successors satisfy q, and in worlds with no
successors, □q holds vacuously. Hence □q is true at all worlds.
Question 5
Which of the following formulas is false at some point in the model?
Answer: □♢p
Justification: At x1 , the accessible world x2 has no successors, so ♢p is false at x2 .
Hence □♢p fails at x1 .
1
, Question 6
Rewrite the English sentence: “It ought to be the case that if it rains outside then it is
permitted to take leave from work” into modal logic using p: ”It rains outside” and q:
”Take leave from work”, with permission formalized as ¬□¬q.
Answer:
□(p → ¬□¬q)
Justification: □ϕ is interpreted as “It ought to be that ϕ.” Permission to q becomes
¬□¬q. Therefore, the conditional obligation becomes □(p → ¬□¬q).
Question 7
Explain why the modal schema □ϕ → □□ϕ corresponds to axiom 4 of modal logic.
Answer: This is axiom 4 of modal logic.
Justification: If ϕ is necessarily true, then it is necessarily necessary. This requires
transitive accessibility relations and ensures that if ϕ holds in all reachable worlds, then
it holds in all worlds reachable from those as well.
Question 8
Under the epistemic interpretation of modal operators (where □ϕ means ”the agent knows
ϕ”), what does the schema □ϕ → □□ϕ express informally?
Answer: If the agent knows ϕ, then the agent knows that they know ϕ.
Justification: This is positive epistemic introspection. It reflects a principle of rational
knowledge that one is aware of their own knowledge.
Question 9
Which of the following modal formulas is not valid in the basic modal system K (nor in
the T–K–4 fragments):
a) □p → p
b) □p ∨ □¬p
c) □p → ♢p
d) ♢p → □♢p
2
